Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model

TL;DR

This paper derives semiclassical asymptotics for orthogonal polynomials with a double-well quartic potential, analyzes their recursive coefficients, and proves eigenvalue universality in the associated matrix model.

Contribution

It introduces new semiclassical asymptotics for orthogonal polynomials with a double-well potential and analyzes the behavior of recursive coefficients in the large n, N limit.

Findings

Recursive coefficients form a period-two cycle that drifts slowly.

Semiclassical asymptotics are derived using integrable systems and Riemann-Hilbert methods.

Universality of eigenvalue distribution in the matrix model is established.

Abstract

We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that \epsilon \le (n/N) \le \lambda_{cr} - \epsilon for some \epsilon > 0, where \lambda_{cr} is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously we derive semiclassical asymptotics for the recursive coefficients of the orthogonal polynomials, and we show that these coefficients form a cycle of period two which drifts slowly with the change of the ratio n/N. The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix Riemann-Hilbert problem. As an application of the semiclassical asymptotics of the orthogonal polynomials, we prove the universality of the local distribution of eigenvalues in the matrix model with the double-well quartic interaction in the presence of two cuts.